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## Q. 2.6.4

Impulse Response of a First-Order Model

Obtain the unit-impulse response of the following model. The initial condition is x(0−)=0. What is the value of x(0+)?

$\frac{X(s)}{F(s)}=\frac{1}{s+5}$

## Verified Solution

Because f(t) = 𝛿(t), F(s) = 1, and the response is obtained from

$X(s)=\frac{1}{s+5}F(s)=\frac{1}{s+5}$

The response is $x(t) = e^{−5t}$ for t > 0. This gives

$x(0+)=\underset{t \rightarrow 0+}{\text{lim}} x(t)= \underset{t \rightarrow 0+}{\text{lim}} e^{-5t}=1$

So the impulse input has changed x from 0 at t = 0− to 1 at t = 0+. This same result could have been obtained from the initial-value theorem:

$x(0+)= \underset{s \rightarrow \infty}{\text{lim}} sX(s)= \underset{s \rightarrow \infty}{\text{lim}} s \frac{1}{s+5}=1$